# Node (physics)

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A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an anti-node, a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes. [1]

## Explanation

Standing waves result when two sinusoidal wave trains of the same frequency are moving in opposite directions in the same space and interfere with each other. [2] They occur when waves are reflected at a boundary, such as sound waves reflected from a wall or electromagnetic waves reflected from the end of a transmission line, and particularly when waves are confined in a resonator at resonance, bouncing back and forth between two boundaries, such as in an organ pipe or guitar string.

In a standing wave the nodes are a series of locations at equally spaced intervals where the wave amplitude (motion) is zero (see animation above). At these points the two waves add with opposite phase and cancel each other out. They occur at intervals of half a wavelength (λ/2). Midway between each pair of nodes are locations where the amplitude is maximum. These are called the antinodes. At these points the two waves add with the same phase and reinforce each other.

In cases where the two opposite wave trains are not the same amplitude, they do not cancel perfectly, so the amplitude of the standing wave at the nodes is not zero but merely a minimum. This occurs when the reflection at the boundary is imperfect. This is indicated by a finite standing wave ratio (SWR), the ratio of the amplitude of the wave at the antinode to the amplitude at the node.

In resonance of a two dimensional surface or membrane, such as a drumhead or vibrating metal plate, the nodes become nodal lines, lines on the surface where the surface is motionless, dividing the surface into separate regions vibrating with opposite phase. These can be made visible by sprinkling sand on the surface, and the intricate patterns of lines resulting are called Chladni figures.

In transmission lines a voltage node is a current antinode, and a voltage antinode is a current node.

Nodes are the points of zero displacement, not the points where two constituent waves intersect.

## Boundary conditions

Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection:

• Fixed boundary: Examples of this type of boundary are the attachment point of a guitar string, the closed end of an open pipe like an organ pipe, or a woodwind pipe, the periphery of a drumhead, a transmission line with the end short circuited, or the mirrors at the ends of a laser cavity. In this type, the amplitude of the wave is forced to zero at the boundary, so there is a node at the boundary, and the other nodes occur at multiples of half a wavelength from it:
0,  λ/2,  λ,  3λ/2,  2λ, ..., nλ/2
• Free boundary: Examples of this type are an open-ended organ or woodwind pipe, the ends of the vibrating resonator bars in a xylophone, glockenspiel or tuning fork, the ends of an antenna, or a transmission line with an open end. In this type the derivative (slope) of the wave's amplitude (in sound waves the pressure, in electromagnetic waves, the current) is forced to zero at the boundary. So there is an amplitude maximum (antinode) at the boundary, the first node occurs a quarter wavelength from the end, and the other nodes are at half wavelength intervals from there:
λ/4,  3λ/4,  5λ/4,  7λ/4, ..., (2n+1)λ/4

## Examples

### Sound

A sound wave consists of alternating cycles of compression and expansion of the wave medium. During compression, the molecules of the medium are forced together, resulting in the increased pressure and density. During expansion the molecules are forced apart, resulting in the decreased pressure and density.

The number of nodes in a specified length is directly proportional to the frequency of the wave.

Occasionally on a guitar, violin, or other stringed instrument, nodes are used to create harmonics. When the finger is placed on top of the string at a certain point, but does not push the string all the way down to the fretboard, a third node is created (in addition to the bridge and nut) and a harmonic is sounded. During normal play when the frets are used, the harmonics are always present, although they are quieter. With the artificial node method, the overtone is louder and the fundamental tone is quieter. If the finger is placed at the midpoint of the string, the first overtone is heard, which is an octave above the fundamental note which would be played, had the harmonic not been sounded. When two additional nodes divide the string into thirds, this creates an octave and a perfect fifth (twelfth). When three additional nodes divide the string into quarters, this creates a double octave. When four additional nodes divide the string into fifths, this creates a double-octave and a major third (17th). The octave, major third and perfect fifth are the three notes present in a major chord.

The characteristic sound that allows the listener to identify a particular instrument is largely due to the relative magnitude of the harmonics created by the instrument.

### Waves in two or three dimensions

In two dimensional standing waves, nodes are curves (often straight lines or circles when displayed on simple geometries.) For example, sand collects along the nodes of a vibrating Chladni plate to indicate regions where the plate is not moving. [3]

In chemistry, quantum mechanical waves, or "orbitals", are used to describe the wave-like properties of electrons. Many of these quantum waves have nodes and antinodes as well. The number and position of these nodes and antinodes give rise to many of the properties of an atom or covalent bond. Atomic orbitals are classified according to the number of radial and angular nodes. A radial node for the hydrogen atom is a sphere that occurs where the wavefunction for an atomic orbital is equal to zero, while the angular node is a flat plane. [4]

Molecular orbitals are classified according to bonding character. Molecular orbitals with an antinode between nuclei are very stable, and are known as "bonding orbitals" which strengthen the bond. In contrast, molecular orbitals with a node between nuclei will not be stable due to electrostatic repulsion and are known as "anti-bonding orbitals" which weaken the bond. Another such quantum mechanical concept is the particle in a box where the number of nodes of the wavefunction can help determine the quantum energy statezero nodes corresponds to the ground state, one node corresponds to the 1st excited state, etc. In general, [5] If one arranges the eigenstates in the order of increasing energies, ${\displaystyle \epsilon _{1},\epsilon _{2},\epsilon _{3},...}$, the eigenfunctions likewise fall in the order of increasing number of nodes; the nth eigenfunction has n−1 nodes, between each of which the following eigenfunctions have at least one node.

## Related Research Articles

In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term atomic orbital may also refer to the physical region or space where the electron can be calculated to be present, as predicted by the particular mathematical form of the orbital.

The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as f0, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f1, the first harmonic.

A harmonic series is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency.

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a traveling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. Waves are often described by a wave equation or a one-way wave equation for single wave propagation in a defined direction.

Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force is equal or close to a natural frequency of the system on which it acts. When an oscillating force is applied at a resonant frequency of a dynamic system, the system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies.

In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes, and the locations where the absolute value of the amplitude is maximum are called antinodes.

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, ions, or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as

A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields.

An amphidromic point, also called a tidal node, is a geographical location which has zero tidal amplitude for one harmonic constituent of the tide. The tidal range for that harmonic constituent increases with distance from this point, though not uniformly. As such, the concept of amphidromic points is crucial to understanding tidal behaviour. The term derives from the Greek words amphi ("around") and dromos ("running"), referring to the rotary tides which circulate around amphidromic points.

A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions.

A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator can be either electromagnetic or mechanical. Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal. Musical instruments use acoustic resonators that produce sound waves of specific tones. Another example is quartz crystals used in electronic devices such as radio transmitters and quartz watches to produce oscillations of very precise frequency.

A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos.

An organ pipe is a sound-producing element of the pipe organ that resonates at a specific pitch when pressurized air is driven through it. Each pipe is tuned to a specific note of the musical scale. A set of organ pipes of similar timbre comprising the complete scale is known as a rank; one or more ranks constitutes a stop.

In electronics, a Lecher line or Lecher wires is a pair of parallel wires or rods that were used to measure the wavelength of radio waves, mainly at VHF, UHF and microwave frequencies. They form a short length of balanced transmission line. When attached to a source of radio-frequency power such as a radio transmitter, the radio waves form standing waves along their length. By sliding a conductive bar that bridges the two wires along their length, the length of the waves can be physically measured. Austrian physicist Ernst Lecher, improving on techniques used by Oliver Lodge and Heinrich Hertz, developed this method of measuring wavelength around 1888. Lecher lines were used as frequency measuring devices until frequency counters became available after World War 2. They were also used as components, often called "resonant stubs", in VHF, UHF and microwave radio equipment such as transmitters, radar sets, and television sets, serving as tank circuits, filters, and impedance-matching devices. They are used at frequencies between HF/VHF, where lumped components are used, and UHF/SHF, where resonant cavities are more practical.

Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration.

Kundt's tube is an experimental acoustical apparatus invented in 1866 by German physicist August Kundt for the measurement of the speed of sound in a gas or a solid rod. The experiment is still taught today due to its ability to demonstrate longitudinal waves in a gas. It is used today only for demonstrating standing waves and acoustical forces.

Violin acoustics is an area of study within musical acoustics concerned with how the sound of a violin is created as the result of interactions between its many parts. These acoustic qualities are similar to those of other members of the violin family, such as the viola.

A wind instrument is a musical instrument that contains some type of resonator in which a column of air is set into vibration by the player blowing into a mouthpiece set at or near the end of the resonator. The pitch of the vibration is determined by the length of the tube and by manual modifications of the effective length of the vibrating column of air. In the case of some wind instruments, sound is produced by blowing through a reed; others require buzzing into a metal mouthpiece, while yet others require the player to blow into a hole at an edge, which splits the air column and creates the sound.

## References

1. Stanford, A. L.; Tanner, J. M. (2014). Physics for Students of Science and Engineering. Academic Press. p. 561. ISBN   148322029X.
2. Feynman, Richard P.; Robert Leighton; Matthew Sands (1963). The Feynman Lectures on Physics, Vol.1 . USA: Addison-Wesley. pp. ch.49. ISBN   0-201-02011-4.
3. Comer, J. R., et al. "Chladni plates revisited." American journal of physics 72.10 (2004): 1345-1346.
4. Supplemental modules (physical and Theoretical Chemistry). Chemistry LibreTexts. (2020, December 13). Retrieved September 13, 2022, from https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)
5. Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. online Ch 3  §12